Signals may be used to transmit data over distances. In optical communication systems, for example, data may be modulated on one or more optical wavelengths to produce modulated optical signals that may be transmitted over optical waveguides such as optical fibers. Optical communications systems have used techniques, such as wavelength division multiplexing (WDM) and various multi-level modulation formats, to increase the transmission capacity. In a multi-level modulation format, multiple data bits may be encoded on a single transmitted symbol. Multi-level modulation techniques have been used, for example, to allow increased transmission rates and decreased channel spacing, thereby increasing the spectral efficiency (SE) of each channel in a WDM system.
One type of modulation that may be used in optical communication systems is phase shift keying (PSK). According to different variations of PSK, data is transmitted by modulating the phase of an optical wavelength such that the phase or phase transition of the optical wavelength represents symbols encoding one or more bits. In a binary phase-shift keying (BPSK) modulation scheme, for example, two phases may be used to represent 1 bit per symbol. In a quadrature phase-shift keying (QPSK) modulation scheme, four phases may be used to encode 2 bits per symbol. In quadrature amplitude modulation (QAM), information may be modulated using a combination of phase shift keying and amplitude shift keying, for example, to encode multiple bits per symbol. A 16-QAM modulation format may be used, for example, to encode 4 bits per symbol. Other phase shift keying formats include differential phase shift keying (DPSK) formats and variations of phase shift keying and differential phase shift keying formats, such as return-to-zero DPSK (RZ-DPSK) and phase division multiplexed QPSK (PDM-QPSK).
In one example, a PDM-QPSK modulation format has been used successfully as a modulation format for 100 Gb/s transmission over 50 GHz channel spacing, i.e., a 200% SE. Reducing the channel spacing to 25 GHz or less would further improve the spectral efficiency (e.g., to 400% SE or higher) for a 100 Gb/s transmission. To avoid crosstalk from neighboring channels with such a reduction in channel spacing, the modulated optical signal should be passed through a narrow optical filter. One example of a higher multi-level modulation format capable of accomplishing these higher transmission rates with 400% SE is a 16-QAM modulation format. The 16-QAM modulation format, however, has a lower sensitivity (e.g. 4 dB less) than QPSK modulation format and is more sensitive to optical phase noise. Thus, 16-QAM with 400% SE has a limited transmission distance (e.g., around 1000 km).
Using a pre-filtered PDM-QPSK signal and narrow optical filters in an attempt to achieve these higher transmission rates with 400% SE is possible but presents unique challenges in the detection of the received PDM-QPSK signal. In some optical communications systems, the receivers include maximum-likelihood sequence estimation (MLSE) detectors using a Viterbi algorithm to detect and decode the modulated optical signals by determining the most probable input data sequence for the received signals. Long-length inter-symbol interference (ISI) generated by narrow optical filtering of the PDM-QPSK signal may prevent the implementation of the MLSE detector in real time. One attempt to reduce the ISI length by passing the QPSK signal through a one bit delay interferometer is described in co-pending U.S. patent application Ser. No. 12/905,717 filed on Oct. 15, 2010, which is fully incorporated herein by reference. The quadrature-duobinary (QDB)-QPSK signal generated using this technique has an ambiguity in the signal constellation, however, and thus caused an error event length longer than 2 symbols, which may increase bit error rate (BER).